2 SPACES OF BOUNDED CONTINUOUS FUNCTIONS 135
is isometric to «£(E) (resp #£(£)). In particular, the real normed space
underlying #{?(£) is the topological direct sum #£(£) + ffig(E).
(7.2.1) 2Tte subspace #£(E) w cfoyerf fa #F(E); 7/2 o//?£r words, a uniform
limit of bounded continuous functions is continuous.
Indeed, let (/„) be a sequence of bounded continuous mappings of E
into F, which converges to g in #F(E); for any & > 0, there is therefore an
integer n0 such that \\fn - g\\ < e/3 for n^n0. For any t0 e E, let V be a
neighborhood of t0 such that \\fno(t) -/«0(r0)|| < e/3 for any t e V. Then, as
ll/»o(0 - 0(0 II < e/3 for any t e E, we have \\g(t) - g(tQ)\ < e for any t e V,
which proves the continuity of g.
Well-known examples (e.g. the functions x-+xn in [0, 1]) show that a
limit of a simply convergent sequence of continuous functions need not be
continuous. On the other hand, examples are easily given of sequences of
continuous functions which converge nonuniformly to a continuous function
(see Problem 2). However (see also (7.5.6)):
(7.2.2) (Dini's theorem) Let E be a compact metric space. If an increasing
(resp. decreasing) sequence (/„) of real-valued continuous functions converges
simply to a continuous function g, it converges uniformly to g.
Suppose the sequence is increasing. For each £ > 0 and each t e E;
there is an index n(t) such that for m ^ n(t), g(t) —fm(t) < e/3. As g and
/n(r) are continuous, there is a neighborhood V(/) of t such that the relation
feV(t) implies \g(t)-g(t')\^el3 and |/n(t)(0 -/n(0(OI ^ fi/3; hence,
for any t' e V(0 we have g(tf) -fn(t)(t') < e. Take now a finite number
of points f, in E such that the V(tt) cover E, and let nQ be the largest of
the integers n(tt). Then for any t e E, / belongs to one of the V(tt)9 hence,
for n&n09 g(t) -fn(t) ^ g(f) -fno(t) ^ g(f) -fn(ti}(t) < e. Q.E.D.
PROBLEMS
1. Let E be a metric space, F a normed space, (un) a sequence of bounded continuous
mappings of E into F which converges simply in E to a bounded function v.
(a) In order that v be continuous at a point *o e E, it is necessary and sufficient that
for any e > 0 and any integer m, there exist a neighborhood V of *0 and an index n > m
such that \\v(x) — WH(JC)|| =^ e for every x e V.
(b) Suppose in addition E is compact. Then, in order that v be continuous in E, it is
necessary and sufficient that for any e > 0 and any integer m, there exist a finite number